Maharashtra State Board Class 12 (HSC Board) Commerce Mathematics & Statistics Solutions Chapter 1 Mathematical Logic Miscellaneous Exercise 1 - Digest Answers | Maharashtra Board Solutions

Balbharati Maharashtra State Board12th Commerce Maths Solution Book PdfChapter 1 Mathematical Logic Miscellaneous Exercise 1 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 1 Mathematical Logic Miscellaneous Exercise 1

(I) Choose the correct alternative:

Question 1 Maharashtra Board Solution

Which of the following is not a statement? (a) Smoking is injurious to health (b) 2 + 2 = 4 (c) 2 is only even prime number (d) Come here

Solution & Step-by-Step Answer:

(d) Come here

Question 2 Maharashtra Board Solution

Which of the following is an open statement? (a) x is a natural number (b) Give me a glass of water (c) Wish you best of luck (d) Good morning to all

Solution & Step-by-Step Answer:

(a) x is a natural number

Question 3 Maharashtra Board Solution

Let p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r). Then this law is known as (a) Commutative law (b) Associative law. (c) De Morgan’s law (d) Distributive law

Solution & Step-by-Step Answer:

(d) Distributive law

Question 4 Maharashtra Board Solution

The false statement in the following is: (a) p ∧ (~p) is a contradiction (b) (p → q) ↔ (~q → ~p) is a contradiction (c) ~(~p) ↔ p is a tautology (d) p ∨ (~p) ↔ p is a tautology.

Solution & Step-by-Step Answer:

(b) (p → q) ↔ (~q → ~p) is a contradiction

Question 5 Maharashtra Board Solution

Consider the following three statements p : 2 is an even number. q : 2 is a prime number. r : Sum of two prime numbers is always even. Then, the symbolic statement (p ∧ q) → ~r means: (a) 2 is an even and prime number and the sum of two prime numbers is always even. (b) 2 is an even and prime number and the sum of two prime numbers is not always even. (c) If 2 is an even and prime number, then the sum of two prime numbers is not always even. (d) If 2 is an even and prime number, then the sum of two prime numbers is also even.

Solution & Step-by-Step Answer:

Question 6 Maharashtra Board Solution

If p : He is intelligent. q : He is strong. Then, symbolic form of statement: ‘It is wrong that, he is intelligent or strong’ is (a) ~p ∨ ~p (b) ~(p ∧ q) (c) ~(p ∨ q) (d) p ∨ ~q

Solution & Step-by-Step Answer:

Question 7 Maharashtra Board Solution

The negation of the proposition ‘If 2 is prime, then 3 is odd’, is (a) If 2 is not prime, then 3 is not odd (b) 2 is prime and 3 is not odd (c) 2 is not prime and 3 is odd (d) If 2 is not prime, then 3 is odd

Solution & Step-by-Step Answer:

(b) 2 is prime and 3 is not odd

Question 8 Maharashtra Board Solution

The statement (~p ∧ q) ∨ ~q is (a) p ∨ q (b) p ∧ q (c) ~(p ∨ q) (d) ~(p ∧ q)

Solution & Step-by-Step Answer:

(d) ~(p ∧ q) Hint: (~p ∧ q) ∨ ~q = (~p ∨ ~q) ∧ (q ∨ ~q) = (~p ∨ ~q) ∧ t = ~p ∨ ~q = ~(p ∧ q)

Question 9 Maharashtra Board Solution

Which of the following is always true? (a) ~(p → q) ≡ ~q → ~p (b) ~(p ∨ q) ≡ ~p ∨ ~q (c) ~(p → q) ≡ p ∧ ~q (d) ~(p ∧ q) ≡ ~p ∧ ~q

Solution & Step-by-Step Answer:

Question 10 Maharashtra Board Solution

~(p ∨ q) ∨ (~p ∧ q) is logically equivalent to (a) ~p (b) p (c) q (d) ~q

Solution & Step-by-Step Answer:

(a) ~p Hint: ~(p ∨ q) ∨ (~p ∧ q) ≡ (~p ∧ ~q) ∨ (~p ∧ q) ≡ ~p ∧ (~q ∨ q) ≡ ~p ∧ t ≡ ~p

Question 11 Maharashtra Board Solution

If p and q are two statements, then (p → q) ↔ (~q → ~p) is (a) contradiction (b) tautology (c) neither (a) nor (b) (d) none of these

Solution & Step-by-Step Answer:

(b) tautology

Question 12 Maharashtra Board Solution

If p is the sentence ‘This statement is false’, then (a) truth value of p is T (b) truth value of p is F (c) p is both true and false (d) p is neither true nor false

Solution & Step-by-Step Answer:

(d) p is neither true nor false

Question 13 Maharashtra Board Solution

Conditional p → q is equivalent to (a) p → ~q (b) ~p ∨ q (c) ~p → ~q (d) p ∨ ~q

Solution & Step-by-Step Answer:

(b) ~p ∨ q

Question 14 Maharashtra Board Solution

Negation of the statement ‘This is false or That is true’ is (a) That is true or This is false (b) That is true and This is false (c) This is true and That is false (d) That is false and That is true

Solution & Step-by-Step Answer:

Question 15 Maharashtra Board Solution

If p is any statement, then (p ∨ ~p) is a (a) contingency (b) contradiction (c) tautology (d) none of them

Solution & Step-by-Step Answer:

(II) Fill in the blanks:

Question 1 Maharashtra Board Solution

The statement q → p is called as the ___________ of the statement p → q.

Solution & Step-by-Step Answer:

Converse

Question 2 Maharashtra Board Solution

Conjunction of two statements p and q is symbolically written as

Solution & Step-by-Step Answer:

p ∧ q

Question 3 Maharashtra Board Solution

If p ∨ q is true, then truth value of ~p ∨ ~q is ___________

Solution & Step-by-Step Answer:

False

Question 4 Maharashtra Board Solution

Negation of ‘some men are animal’ is ___________

Solution & Step-by-Step Answer:

All men are not animal. OR No men are animals.

Question 5 Maharashtra Board Solution

Truth value of if x = 2, then x2 = -4 is ___________

Solution & Step-by-Step Answer:

False

Question 6 Maharashtra Board Solution

Inverse of statement pattern p → q is given by ___________

Solution & Step-by-Step Answer:

~p → ~q

Question 7 Maharashtra Board Solution

p ↔ q is false when p and q have ___________ truth values.

Solution & Step-by-Step Answer:

Different

Question 8 Maharashtra Board Solution

Let p : The problem is easy. r : It is not challenging. Then verbal form of ~p → r is ___________

Solution & Step-by-Step Answer:

If the problem is not easy, then it is not challenging.

Question 9 Maharashtra Board Solution

Truth value of 2 + 3 = 5 if and only if -3 > -9 is ___________

Solution & Step-by-Step Answer:

T [Hint: T ↔ T = T]

(III) State whether each of the following is True or False:

Question 1 Maharashtra Board Solution

Truth value of 2 + 3 < 6 is F.

Solution & Step-by-Step Answer:

False

Question 2 Maharashtra Board Solution

There are 24 months in a year is a statement.

Solution & Step-by-Step Answer:

True

Question 3 Maharashtra Board Solution

p ∧ q has truth value F if both p and q have truth value F.

Solution & Step-by-Step Answer:

False

Question 4 Maharashtra Board Solution

The negation of 10 + 20 = 30 is, it is false that 10 + 20 ≠ 30.

Solution & Step-by-Step Answer:

False

Question 5 Maharashtra Board Solution

Dual of (p ∧ ~q) ∨ t is (p ∨ ~q) ∨ c.

Solution & Step-by-Step Answer:

False

Question 6 Maharashtra Board Solution

Dual of ‘John and Ayub went to the forest’ is ‘John or Ayub went to the forest.’

Solution & Step-by-Step Answer:

True

Question 7 Maharashtra Board Solution

‘His birthday is on 29th February’ is not a statement.

Solution & Step-by-Step Answer:

True

Question 8 Maharashtra Board Solution

x2 = 25 is true statement.

Solution & Step-by-Step Answer:

False

Question 9 Maharashtra Board Solution

The truth value of ‘√5 is not an irrational number’ is T.

Solution & Step-by-Step Answer:

False

Question 10 Maharashtra Board Solution

p ∧ t = p.

Solution & Step-by-Step Answer:

True

(IV) Solve the following:

Question 1 Maharashtra Board Solution

State which of the following sentences are statements in logic: (i) Ice cream Sundaes are my favourite.

Solution & Step-by-Step Answer:

It is a statement.

(ii) x + 3 = 8, x is variable.
Solution:
It is a statement.

(iii) Read a lot to improve your writing skill.
Solution:
It is an imperative sentence, hence it is not a statement.

(iv) z is a positive number.
Solution:
It is an open sentence, hence it is not a statement.

(v) (a + b)²= a²+ 2ab + b²for all a, b ∈ R.
Solution:
It is a statement.

(vi) (2 + 1)²= 9.
Solution:
It is a statement.

(vii) Why are you sad?
Solution:
It is an interrogative sentence, hence it is not a statement.

(viii) How beautiful the flower is!
Solution:
It is an exclamatory sentence, hence it is not a statement.

(ix) The square of any odd number is even.
Solution:
It is a statement.

(x) All integers are natural numbers.
Solution:
It is a statement.

(xi) If x is a real number, then x²≥ 0.
Solution:
It is a statement.

(xii) Do not come inside the room.
Solution:
It is an imperative sentence, hence it is not a statement.

(xiii) What a horrible sight it was!
Solution:
It is an exclamatory sentence, hence it is not a statement.

Question 2 Maharashtra Board Solution

Which of the following sentences are statements? In case of a statement, write down the truth value: (i) What is a happy ending?

Solution & Step-by-Step Answer:

It is an interrogative sentence, hence it is not a statement.

(ii) The square of every real number is positive.
Solution:
It is a statement that is false, hence its truth value is F.

(iii) Every parallelogram is a rhombus.
Solution:
It is a statement that is true, hence its truth value is T.

(iv) a²– b²= (a + b)(a – b) for all a, b ∈ R.
Solution:
It is a mathematical identity that is true, hence its truth value is T.

(v) Please carry out my instruction.
Solution:
It is an imperative sentence, hence it is not a statement.

(vi) The Himalayas is the highest mountain range.
Solution:
It is a statement that is true, hence its truth value is T.

(vii) (x – 2)(x – 3) = x²– 5x + 6 for all x ∈ R.
Solution:
It is a mathematical identity that is true, hence its truth value is T.

(viii) What are the causes of rural unemployment?
Solution:
It is an interrogative sentence, hence it is not a statement.

(ix) 0! = 1.
Solution:
It is a statement that is true, hence its truth value is T.

(x) The quadratic equation ax²+ bx + c = 0 (a ≠ 0) always has two real roots.
Solution:
It is a statement that is false, hence its truth value is F.

Question 3 Maharashtra Board Solution

Assuming the first statement as p and second as q, write the following statements in symbolic form: (i) The Sun has set and Moon has risen.

Solution & Step-by-Step Answer:

Let p : The Sun has set. q : Moon has risen. Then the symbolic form of the given statement is p ∧ q.

(ii) Mona likes Mathematics and Physics.
Solution:
Let p : Mona likes Mathematics.
q : Mona likes Physics.
Then the symbolic form of the given statement is p ∧ q.

(iii) 3 is a prime number if 3 is a perfect square number.
Solution:
Let p : 3 be a prime number.
q : 3 is a perfect square number.
Then the symbolic form of the given statement is p ↔ q.

(iv) Kavita is brilliant and brave.
Solution:
Let p : Kavita is brilliant.
q : Kavita is brave.
Then the symbolic form of the given statement is p ∧ q.

(v) If Kiran drives a car, then Sameer will walk.
Solution:
Let p : Kiran drives a car.
q : Sameet will walk.
Then the symbolic form of the given statement is p → q.

(vi) The necessary condition for the existence of a tangent to the curve of the function is continuity.
Solution:
The given statement can be written as:
‘If the function is continuous, then the tangent to the curve exists.’
Let p : The function is continuous.
q : Tangent to the curve exists.
Then the symbolic form of the given statement is p → q.

(vii) To be brave is necessary and sufficient condition to climb Mount Everest.
Solution:
Let p : To be brave.
q : Climb Mount Everest.
Then the symbolic form of the given statement is p ↔ q.

(viii) x³+ y³= (x + y)³, iff xy = 0.
Solution:
Let p : x³+ y³= (x + y)³.
q : xy = 0.
Then the symbolic form of the given statement is p ↔ q.

(ix) The drug is effective though it has side effects.
Solution:
Let p : The drug is effective.
q : It has side effects.
Then the symbolic form of the given statement is p ∧ q.

(x) If a real number is not rational, then it must be irrational.
Solution:
Let p : A real number is not rational.
q : It must be irrational.
Then the symbolic form of the given statement is p → q.

(xi) It is not true that Ram is tall and handsome.
Solution:
Let p : Ram is tall.
q : Ram is handsome.
Then the symbolic form of the given statement is ~(p ∧ q).

(xii) Even though it is not cloudy, it is still raining.
Solution:
The given statement is equivalent to:
It is not cloudy and it is still raining,
Let p : It is not cloudy.
q : It is still raining.
Then the symbolic form of the given statement is p ∧ q.

(xiii) It is not true that intelligent persons are neither polite nor helpful.
Solution:
Let p : Intelligent persons are neither polite nor helpful.
Then the symbolic form of the given statement is ~p.

(xiv) If the question paper is not easy, then we shall not pass.
Solution:
Let p : The question paper is not easy.
q : We shall not pass.
Then the symbolic form of the given statement is p → q.

Question 4 Maharashtra Board Solution

If p : Proof is lengthy. q : It is interesting. Express the following statements in symbolic form: (i) Proof is lengthy and it is not interesting. (ii) If the proof is lengthy, then it is interesting. (iii) It is not true that the proof is lengthy but it is interesting. (iv) It is interesting iff the proof is lengthy.

Solution & Step-by-Step Answer:

The symbolic form of the given statements are: (i) p ∧ ~q (ii) p → q (iii) ~(p ∧ q) (iv) q ↔ p

Question 5 Maharashtra Board Solution

Let p : Sachin win the match. q : Sachin is a member of the Rajya Sabha. r : Sachin is happy. Write the verbal statement for each of the following: (i) (p ∧ q) ∨ r

Solution & Step-by-Step Answer:

Sachin wins the match and he is a member of the Rajya Sabha or Sachin is happy.

(ii) p → r
Solution:
If Sachin wins the match, then he is happy.

(iii) ~p ∨ q
Solution:
Sachin does not win the match or he is a member of the Rajya Sabha.

(iv) p → (q ∨ r)
Solution:
If Sachin wins the match, then he is a member of the Rajya Sabha or he is happy.

(v) p → q
Solution:
If Sachin wins the match, then he is a member of the Rajya Sabha.

(vi) (p ∧ q) ∧ ~r
Solution:
Sachin wins the match and he is a member of the Rajya Sabha but he is not happy.

(vii) ~(p ∨ q) ∧ r
Solution:
It is false that Sachin wins the match or he is a member of the Rajya Sabha but he is happy.

Question 6 Maharashtra Board Solution

Determine the truth values of the following statements: (i) 4 + 5 = 7 or 9 – 2 = 5.

Solution & Step-by-Step Answer:

Let p : 4 + 5 = 7. q : 9 – 2 = 5. Then the symbolic form of the given statement is p ∨ q. The truth values of both p and q are F. ∴ the truth value of p ∨ q is F. …….[F ∨ F ≡ F]

(ii) If 9 > 1, then x²– 2x + 1 = 0 for x = 1.
Solution:
Let p : 9 > 1.
q : x²– 2x + 1 = 0 for x = 1.
Then the symbolic form of the given statement is p → q.
The truth values of both p and q are T.
∴ the truth value of p → q is T. …..[T → T ≡ T]

(iii) x + y = 0 is the equation of a straight line if and only if y²= 4x is the equation of the parabola.
Solution:
Let p : x + y = 0 is the equation of a straight line.
q : y²= 4x is the equation of the parabola.
Then the symbolic form of the given statement is p ↔ q.
The truth values of both p and q are T.
∴ the truth value of p ↔ q is T. …..[T ↔ T ≡ T]

(iv) It is not true that 2 + 3 = 6 or 12 + 3 = 5.
Solution:
Let p : 2 + 3 = 6.
q : 12 + 3 = 5.
Then the symbolic form of the given statement is ~(p ∨ q).
The truth values of both p and q are F.
∴ the truth value of ~(p ∨ q) is T. …..[~(F ∨ F) ≡ ~F ≡ T]

Question 7 Maharashtra Board Solution

Assuming the following statements p : Stock prices are high. q : Stocks are rising. to be true, find the truth values of the following:

(i) Stock prices are not high or stocks are rising.

Solution & Step-by-Step Answer:

p and q are true, i.e. T.
∴ ~p and ~q are false, i.e. F.
The given statement in symbolic form is ~p ∨ q.
Since, ~T ∨ T ≡ F ∨ T ≡ T, the given statement is true.
Hence, its truth value is ‘T’.

(ii) Stock prices are high and stocks are rising if and only if stock prices are high.
Solution:
The given statement in symbolic form is (p ∧ q) ↔ p.
Since (T ∧ T) ↔ T ≡ T ↔ T ≡ T, the given statement is true.
Hence, its truth value is ‘T’.

(iii) If stock prices are high, then stocks are not rising.
Solution:
The given statement in symbolic form is p → ~q.
Since, T → ~T ≡ T → F ≡ F, the given statement is false.
Hence, its truth value is ‘F’.

(iv) It is false that stocks are rising and stock prices are high.
Solution:
The given statement in symbolic form is ~(q ∧ p).
Since, ~(T ∧ T) ≡ ~T ≡ F, the given statement is false.
Hence, its truth value is ‘F’.

(v) Stock prices are high or stocks are not rising iff stocks are rising.
Solution:
The given statement in symbolic form is (p ∨ ~q) ↔ q.
Since (T ∨ ~T) ↔ T ≡ (T ∨ F) ↔ T
≡ T ↔ T
≡ T, the given statement is true.
Hence, its truth value is ‘T’.

Question 8 Maharashtra Board Solution

Rewrite the following statements without using conditional: [Hint: P → q ≡ ~p ∨ q] (i) If price increases, then demand falls. (ii) If demand falls, then the price does not increase.

Solution & Step-by-Step Answer:

Since, p → q ≡ ~p ∨ q, the given statements can be written as: (i) Price does not increase or demand falls. (ii) Demand does not fall or price does not increase.

Question 9 Maharashtra Board Solution

If p, q, r are statements with truth values T, T, F respectively, determine the truth values of the following: (i) (p ∧ q) → ~p

Solution & Step-by-Step Answer:

Truth values of p, q, r are T, T, F respectively. (p ∧ q) → ~p ≡ (T ∧ T) → ~T ≡ T → F ≡ F Hence, the truth value of the given statement is false, i.e. F.

(ii) p ↔ (q → ~p)
Solution:
p ↔ (q → ~p) ≡ T ↔ (T → ~T)
≡ T ↔ (T → F)
≡ T ↔ F
≡ F
Hence, the truth value of the given statement is false, i.e. F.

(iii) (p ∧ ~q) ∨ (~p ∧ q)
Solution:
(p ∧ ~q) ∨ (~p ∧ q) ≡ (T ∧ ~T) ∨ (~T ∧ T)
≡ (T ∧ F) ∨ (F ∧ T)
≡ F ∨ F
≡ F
Hence, the truth value of the given statement is false, i.e. F.

(iv) ~(p ∧ q) → ~(q ∧ p)
Solution:
~(p ∧ q) → ~(q ∧ p) ≡ ~(T ∧ T) → ~(T ∧ T)
≡ ~T → ~T
≡ F → F
≡ T
Hence, the truth value of the given statement is true, i.e. T.

(v) ~[(p → q) ↔ (p ∧ ~q)]
Solution:
~[(p → q) ↔ (p ∧ ~q)]
≡ ~[(T → T) ↔ (T ∧ ~T)]
≡ ~[T ↔ (T ∧ F)]
≡ ~[T ↔ F]
≡ ~F
≡ T.
Hence, the truth value of the given statement is true, i.e. T.

Question 10 Maharashtra Board Solution

Write the negations of the following: (i) If ΔABC is not equilateral, then it is not equiangular.

Solution & Step-by-Step Answer:

Let p : ΔABC is not equilateral. q : It is not equiangular. Then the symbolic form of the given statement is p → q. Since, ~(p → q) ≡ p ∧ ~q, the negation of the given statement is: ‘ΔABC is not equilateral and it is equiangular.’

(ii) Ramesh is intelligent and he is hard working.
Solution:
Let p : Ramesh is intelligent.
q : He is hard working.
Then the symbolic form of the given statement is p ∧ q.
Since, ~(p ∧ q) ≡ ~p ∨ ~q, the negation of the given statement is:
‘Ramesh is not intelligent or he is not hard-working.’

(iii) A angle is a right angle if and only if it is of measure 90°.
Solution:
Let p : An angle is a right angle.
q : It is of measure 90°.
Then the symbolic form of the given statement is p ↔ q.
Since, ~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p), the negation of the given statement is:
‘An angle is a right angle and it is not of measure 90° or an angle is of measure 90° and it is not a right angle.’

(iv) Kanchanjunga is in India and Everest is in Nepal.
Solution:
Let p : Kanchenjunga is in India.
q : Everest is in Nepal.
Then the symbolic form of the given statement is p ∧ q.
Since, ~(p ∧ q) ≡ ~p ∨ ~q, the negation of the given statement is:
‘Kanchenjunga is not in India or Everest is not in Nepal.’

(v) If x ∈ A ∩ B, then x ∈ A and x ∈ B.
Solution:
Let p : x ∈ A ∩ B, q : x ∈ A, r : x ∈ B.
Then the symbolic form of the given statement is P → (q ∧ r)
Since, ~(p → q) ≡ p ∧ ~q and ~(p ∧ q)= ~p ∨ ~q,
the negation of the given statement is:
‘x ∈ A ∩ B and x ∉ A or x ∉ B.

Question 11 Maharashtra Board Solution

Construct the truth table for each of the following statement patterns: (i) (p ∧ ~q) ↔ (q → p)

Solution & Step-by-Step Answer:

(p ∧ ~q) ↔ (q → p)

(ii) (~p ∨ q) ∧ (~p ∧ ~q)
Solution:
(~p ∨ q) ∧ (~p ∧ ~q)

(iii) (p ∧ r) → (p ∨ ~q)
Solution:
(p ∧ r) → (p ∨ ~q)

(iv) (p ∨ r) → ~(q ∧ r)
Solution:
(p ∨ r) → ~(q ∧ r)

(v) (p ∨ ~q) → (r ∧ p)
Solution:
(p ∨ ~q) → (r ∧ p)

Question 12 Maharashtra Board Solution

What is a tautology? What is a contradiction? Show that the negation of a tautology is a contradiction and the negation of a contradiction is a tautology.

Solution & Step-by-Step Answer:

Tautology: A statement pattern that has all the entries in the last column of its truth table as T is called a tautology. For example: In the above truth table for the statement p ∨ ~p, we observe that all the entries in the last column are T. Hence, the statement p ∨ ~p is a tautology.

Contradiction: A statement pattern that has all the entries in the last column of its truth table as F is called a contradiction.
For example:

In the above truth table for the statement p ∧ ~p,
we observe that all the entries in the last column are F.
Hence, the statement p ∧ ~p is a contradiction.

To show that the negation of a tautology is a contradiction and vice versa:
A tautology is true on every row of its truth table.
Since, ~T = F and ~F = T, when we negate a tautology, the resulting statement is false on every row of its table.
i.e. the negation of tautology is a contradiction.
Similarly, the negation of a contradiction is a tautology.

Question 13 Maharashtra Board Solution

Determine whether the following statement patterns is a tautology or a contradiction or a contingency: (i) [(p ∧ q) ∨ (~p)] ∨ [p ∧ (~q)]

Solution & Step-by-Step Answer:

[(p ∧ q) ∨ (~p)] ∨ [p ∧ (~q)] All the entries in the last column of the above truth table are T. ∴ [(p ∧ q) ∨ (~p)] ∨ [p ∧ (~q)] is a tautology.

(ii) [(~p ∧ q) ∧ (q ∧ r)] ∨ (~q)
Solution:
[(~p ∧ q) ∧ (q ∧ r)] ∨ (~q)

The entries in the last column of the above truth table are neither all T nor all F.
∴ [(~p ∧ q) ∧ (q ∧ r)] ∨ (~q) is a contingency.

(iii) [~(p ∨ q) → p] ↔ [(~p) ∧ (~q)]
Solution:
[~(p ∨ q) → p] ↔ [(~p) ∧ (~q)]

All the entries in the last column of the above truth table are F.
∴ [~(p ∨ q) → p] ↔ [(~p) ∧ (~q)] is a contradiction.

(iv) [~(p ∧ q) → p] ↔ [(~p) ∧ (~q)]
Solution:
[~(p ∧ q) → p] ↔ [(~p) ∧ (~q)]

The entries in the last column of the above truth table are neither all T nor all F.
∴ [~(p ∧ q) → p] ↔ [(~p) ∧ (~q)] is a contingency.

(v) [p → (~q ∨ r)] ↔ ~[p → (q → r)]
Solution:
[p → (~q ∨ r)] ↔ ~[p → (q → r)]

All the entries in the last column of the above truth table are F.
∴ [p → (~q ∨ r)] ↔ ~[p → (q → r)] is a contradiction.

Question 14 Maharashtra Board Solution

Using the truth table, prove the following logical equivalences: (i) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Solution & Step-by-Step Answer:

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) The entries in columns 5 and 8 are identical. ∴ p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

(ii) [~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r
Solution:
[~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r

The entries in columns 3 and 7 are identical.
∴ [~(p ∨ q) ∨ (p ∨ q)] ∧ r ≡ r

(iii) p ∧ (~p ∨ q) ≡ p ∧ q
Solution:
p ∧ (~p ∨ q) ≡ p ∧ q

The entries in columns 5 and 6 are identical.
∴ p ∧ (~p ∨ q) ≡ p ∧ q

(iv) p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)
Solution:
p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)

The entries in columns 3 and 10 are identical.
∴ p ↔ q ≡ ~(p ∧ ~q) ∧ ~(q ∧ ~p)

(v) ~p ∧ q ≡ (p ∨ q) ∧ ~p
Solution:
~p ∧ q ≡ (p ∨ q) ∧ ~p

The entries in columns 4 and 6 are identical.
∴ ~p ∧ q ≡ (p ∨ q) ∧ ~p

Question 15 Maharashtra Board Solution

Write the converse, inverse, contrapositive of the following statements: (i) If 2 + 5 = 10, then 4 + 10 = 20.

Solution & Step-by-Step Answer:

Let p : 2 + 5 = 10. q : 4 + 10 = 20. Then the symbolic form of the given statement is p → q. Converse: q → p is the converse of p → q i.e. If 4 + 10 = 20, then 2 + 5 = 10. Inverse: ~p → ~q is the inverse of p → q i.e. If 2 + 5 ≠ 10, then 4 + 10 ≠ 20. Cotrapositive: ~q → ~p is the contrapositive of p → q, i.e. If 4 +10 ≠ 20, then 2 + 5 ≠ 10.

(ii) If a man is a bachelor, then he is happy.
Solution:
Let p : A man is a bachelor.
q : He is happy.
Then the symbolic form of the given statement is p → q.
Converse: q → p is the converse of p → q
i.e. If a man is happy, then he is a bachelor.
Inverse: ~p → ~q is the inverse of p → q
i.e. If a man is not a bachelor, then he is not happy.
Contrapositive: ~q → ~p is the contrapositive of p → q
i.e., If a man is not happy, then he is not a bachelor.

(iii) If I do not work hard, then I do not prosper.
Solution:
Let p : I do not work hard.
q : I do not prosper.
Then the symbolic form of the given statement is p → q.
Converse: q → p is the converse of p → q
i.e. If I do not prosper, then I do not work hard.
Inverse: ~p → ~q is the inverse of p → q
i.e. If I work hard, then I prosper.
Contrapositive: ~q → ~p is the contrapositive of p → q
i.e. If I prosper, then I work hard.

Question 16 Maharashtra Board Solution

State the dual of each of the following statements by applying the principle of duality: (i) (p ∧ ~q) ∨ (~p ∧ q) ≡ (p ∨ q) ∧ ~(p ∧ q) (ii) p ∨ (q ∨ r) ≡ ~[(p ∧ q) ∨ (r ∨ s)] (iii) 2 is an even number or 9 is a perfect square.

Solution & Step-by-Step Answer:

The duals are given by: (i) (p ∨ ~q) ∧ (~p ∨ q) ≡ (p ∧ q) ∨ ~(p ∨ q) (ii) p ∧ (q ∧ r) ≡ ~[(p ∨ q) ∧ (r ∧ s)] (iii) 2 is an even number and 9 is a perfect square.

Question 17 Maharashtra Board Solution

Rewrite the following statements without using the connective ‘If … then’: (i) If a quadrilateral is a rhombus, then it is not a square. (ii) If 10 – 3 = 7, then 10 × 3 ≠ 30. (iii) If it rains, then the principal declares a holiday.

Solution & Step-by-Step Answer:

Since, p → q ≡ ~p ∨ q the given statements can be written as: (i) A quadrilateral is not a rhombus or it is not a square. (ii) 10 – 3 ≠ 7 or 10 × 3 ≠ 30. (iii) It does not rain or the principal declares a holiday.

Question 18 Maharashtra Board Solution

Write the dual of each of the following: (i) (~p ∧ q) ∨ (p ∧ ~q) ∨ (~p ∧ ~q) (ii) (p ∧ q) ∧ r ≡ p ∧ (q ∧ r) (iii) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) (iv) ~(p ∨ q) ≡ ~p ∧ ~q.

Solution & Step-by-Step Answer:

The duals are given by: (i) (~p ∨ q) ∧ (p ∨ ~q) ∧ (~p ∨ ~q) (ii) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r) (iii) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) (iv) ~(p ∧ q) ≡ ~p ∧ ~q

Question 19 Maharashtra Board Solution

Consider the following statements: (i) If D is a dog, then D is very good. (ii) If D is very good, then D is a dog. (iii) If D is not very good, then D is not a dog. (iv) If D is not a dog, then D is not very good. Identify the pairs of statements having the same meaning. Justify.

Solution & Step-by-Step Answer:

Let p : D is a dog. and q : D is very good. Then the given statements in the symbolic form are: (i) p → q (ii) q → p (iii) ~q → ~p (iv) ~p → ~q The entries in columns (i) and (iii) are identical. Hence, these statements are equivalent. ∴ the statements (i) and (iii) have the same meaning. Similarly, the entries in columns (ii) and (iv) are identical. Hence, these statements are equivalent. ∴ the statements (ii) and (iv) have the same meaning.

Question 20 Maharashtra Board Solution

Express the truth of each of the following statements by Venn diagrams: (i) All men are mortal.

Solution & Step-by-Step Answer:

Let U : a set of all human being A : set of all men B : set of all mortals. Then the Venn diagram represents the truth of the given statement is as below:

(ii) Some persons are not politicians.
Solution:
Let U : set of all human being
A : set of all persons
B : set of all politicians.
Then the Venn diagram represents the truth of the given statement is as follows:

(iii) Some members of the present Indian cricket are not committed.
Solution:
Let U : set of all human being
X : set of all members of present Indian cricket
Y : set of all committed members of the present Indian cricket.
Then the Venn diagram represents the truth of the given statement is as below:

(iv) No child is an adult.
Solution:
Let U : set of all human beings
C : set of all children
A : set of all adults.
Then the Venn diagram represents the truth of the given statement is as below:

Question 21 Maharashtra Board Solution

If A = {2, 3, 4, 5, 6, 7, 8}, determine the truth value of each of the following statements: (i) ∃ x ∈ A, such that 3x + 2 > 9.

Solution & Step-by-Step Answer:

Clearly x = 3, 4, 5, 6, 7, 8 ∈ A satisfy 3x + 2 > 9. So, the given statement is true, hence its truth value is T.

(ii) ∀x ∈ A, x²< 18.
Solution:
x = 5, 6, 7, 8 ∈ A do not satisfy x²< 18.
So the given statement is false, hence its truth value is F.

(iii) ∃x ∈ A, such that x + 3 < 11.
Solution:
Clearly x = 2, 3, 4, 5, 6, 7 ∈ A which satisfy x + 3 < 11.
So, the given statement is True, hence its truth value is T.

(iv) ∀x ∈ A, x²+ 2 ≥ 5.
Solution:
x²+ 2 ≥ 5 for all x ∈ A.
So, the given statement is true, hence its truth value is T.

Question 22 Maharashtra Board Solution

Write the negations of the following statements: (i) 7 is a prime number and the Taj Mahal is in Agra.

Solution & Step-by-Step Answer:

Let p : 7 be a prime number. q : Taj Mahal is in Agra. Then the symbolic form of the given statement is p ∧ q. Since, (p ∧ q) ≡ ~p ∨ ~q, the negation of the given statement is: ‘7 is not a prime number or Taj Mahal is not in Agra.’

(ii) 10 > 5 and 3 < 8.
Solution:
Let p : 10 > 5.
q : 3 < 8.
Then the symbolic form of the given statement is P ∧ q.
Since, ~(p ∧ q) = ~p ∨ ~q, the negation of the given statement is:
’10 ≤ 5 or 3 ≥ 8′
OR
’10 ≯ 5 or 3 ≮ 8′

(iii) I will have tea or coffee.
Solution:
The negation of the given statement is:
‘I will not have tea and coffee.’

(iv) ∀n ∈ N, n + 3 > 9.
Solution:
The negation of the given statement is:
‘∃n ∈ N, such that n + 3 ≯ 9.’
OR
‘∃n ∈ N, such that n + 3 ≤ 9.’

(v) ∃x ∈ A, such that x + 5 < 11.
Solution:
The negation of the given statement is:
‘∀x ∈ A, x + 5 ≮ 1.’
OR
‘∀x ∈ A, x + 5 ≥ 11.’

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Chapter 1 Mathematical Logic Miscellaneous Exercise 1 Solutions